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Properties

Label 2-200-40.3-c1-0-6
Degree $2$
Conductor $200$
Sign $-0.386 - 0.922i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 + 1.26i)2-s + (1.61 + 1.61i)3-s + (−1.17 + 1.61i)4-s + (−1.00 + 3.07i)6-s + (−1.17 − 1.17i)7-s + (−2.79 − 0.442i)8-s + 2.23i·9-s + 1.23·11-s + (−4.52 + 0.715i)12-s + (3.07 − 3.07i)13-s + (0.726 − 2.23i)14-s + (−1.23 − 3.80i)16-s + (1 − i)17-s + (−2.81 + 1.43i)18-s + 2i·19-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)2-s + (0.934 + 0.934i)3-s + (−0.587 + 0.809i)4-s + (−0.408 + 1.25i)6-s + (−0.444 − 0.444i)7-s + (−0.987 − 0.156i)8-s + 0.745i·9-s + 0.372·11-s + (−1.30 + 0.206i)12-s + (0.853 − 0.853i)13-s + (0.194 − 0.597i)14-s + (−0.309 − 0.951i)16-s + (0.242 − 0.242i)17-s + (−0.664 + 0.338i)18-s + 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974641 + 1.46466i\)
\(L(\frac12)\) \(\approx\) \(0.974641 + 1.46466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 1.26i)T \)
5 \( 1 \)
good3 \( 1 + (-1.61 - 1.61i)T + 3iT^{2} \)
7 \( 1 + (1.17 + 1.17i)T + 7iT^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (2.62 - 2.62i)T - 23iT^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 - 5.25iT - 31T^{2} \)
37 \( 1 + (3.07 + 3.07i)T + 37iT^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 + (2.38 + 2.38i)T + 43iT^{2} \)
47 \( 1 + (-7.33 - 7.33i)T + 47iT^{2} \)
53 \( 1 + (-0.726 + 0.726i)T - 53iT^{2} \)
59 \( 1 + 8.47iT - 59T^{2} \)
61 \( 1 + 9.95iT - 61T^{2} \)
67 \( 1 + (-2.38 + 2.38i)T - 67iT^{2} \)
71 \( 1 - 7.05iT - 71T^{2} \)
73 \( 1 + (8.70 + 8.70i)T + 73iT^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (-4.38 - 4.38i)T + 83iT^{2} \)
89 \( 1 - 6.47iT - 89T^{2} \)
97 \( 1 + (-0.236 + 0.236i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.14118568901785357071268792624, −12.04424440920014949026648973650, −10.51508859910807288510723292403, −9.585249148302356130978990063359, −8.689937146364042069446578005398, −7.83153369410343967377575543465, −6.56209052949170425186910484272, −5.30573431360074854962504127005, −3.89169995707503292765480362264, −3.29703875223300126537358009325, 1.66293601948795390577150572353, 2.87171723637115852019108995166, 4.13305227326805171179908027124, 5.87566125278575581001131179873, 6.93090738906352670405269315156, 8.471753333002247427675441591057, 9.061769314599030876701383289156, 10.22349138514401259008603522991, 11.50646537463926710335399239866, 12.25341261954940438983462748149

Graph of the $Z$-function along the critical line